Estimation of ratio of variance components and optimal diallel cross designs

The results on optimal designs for diallel crosses are based on standard linear model assumptions where the general combining ability effects are taken as fixed. Recently Ghosh and Das (2003) proposed a random effects model that allows one to first estimate the variance components and then obtain the variances of these estimates. In this paper we first propose an unbiased estimate of the ratio of the variance components which has a one-to-one relation with heritability. We then obtain a large sample expression of the variance of this unbiased estimator of the ratio of variance components. Through a simulation study, it is shown that even for small samples, the large sample variance is close to the exact variance. Through minimization of the large sample variance we obtain optimal designs and show certain connections with the optimization problem under the fixed effects model.